Editing Adjoint functors (section)

==== Probability ====
The twin fact in probability can be understood as an adjunction: that expectation commutes with affine transform, and that the expectation is in some sense the best ''solution'' to the problem of finding a real-valued approximation to a distribution on the real numbers.

Define a category based on <math>\R</math>, with objects being the real numbers, and the morphisms being "affine functions evaluated at a point". That is, for any affine function <math>f(x) = ax + b</math> and any real number <math>r</math>, define a morphism <math>(r, f): r \to f(r)</math>.

Define a category based on <math>M(\R)</math>, the set of probability distribution on <math>\R</math> with finite expectation. Define morphisms on <math>M(\R)</math> as "affine functions evaluated at a distribution". That is, for any affine function <math>f(x) = ax + b</math> and any <math>\mu\in M(\R)</math>, define a morphism <math>(\mu, f): \mu \to \mu\circ f^{-1}</math>.

Then, the [[Dirac delta measure]] defines a functor: <math>\delta: x\mapsto \delta_x</math>, and the expectation defines another functor <math>\mathbb E: \mu \mapsto \mathbb E[\mu]</math>, and they are adjoint: <math>\mathbb E \dashv \delta</math>. (Somewhat disconcertingly, <math>\mathbb E</math> is the left adjoint, even though <math>\mathbb E</math> is "forgetful" and <math>\delta</math> is "free".)